De Minaur lost out 6-4 6-4 in the US and was on the back foot again for much of the first two sets in Monte Carlo on Tuesday but he toughed it out to win 7-6 (7-5) 2-6 6-2.
First, Norrie, who reached the quarter-finals at Indian Wells before losing to Carlos Alcaraz, edged 4-2 ahead in the opening set but De Minaur clawed him back to win the tiebreaker.
Norrie then won five games in a row en route to forcing a deciding set.
But De Minaur, the No.5 seed, broke serve twice at the start of the final set before progressing to set up a last-16 meeting with either 10th seed Flavio Cobolli or Belgian qualifier Alexander Blockx.
Earlier, fellow Australian Alexei Popyrin was beaten 6-3 6-4 by Casper Ruud, the No.9 seed.
Meanwhile, following first round byes, Jannik Sinner and  Alcaraz opened their clay court seasons with commanding victories as they continue to tussle for the world number one spot.
Sinner defeated Frenchman Ugo Humbert 6-3 6-0 in just 64 minutes, while defending champion Alcaraz overcame Argentina's Sebastian Baez 6-1 6-3.
Yet Alcaraz was quick to point out on Court Rainier III that he expects to lose top spot to Sinner at any time.
"To be honest, I'm going to lose No. 1 in the world [at some point]. I don't know if it is going to be at this tournament or in the next one," said Alcaraz.
"I'm defending a bunch of points that are going to be really difficult to defend. Even if I defend them, Jannik is going to add some points at these tournaments where he doesn't have to defend any points.
"I will try to play my best and let's see what happens, but for me the No. 1 spot is not in my mind right now. I'm just trying to feel the best way on clay courts and let's see how the clay swing goes."
There were also wins for Italy's Matteo Berrettini, No.8 seed Alexander Bublik, Marin Cilic and Tomas Etcheverry.
With agencies
